Pitfalls, Roles and Beyond

[Keith]

Discuss connections you see among the roles of proof and the pitfalls
of dynamic geometry discussed by de Villiers. Also consider
connections between these pitfalls and roles and other readings we
have done thus far in the course.

[CJ]

Let me just first say that I really liked the article on new meanings
for proof. I talked to my 306 students about the Halmos quote on the
second page, and we had a hard time swallowing his statement that
"Efficiency is meaningless." One student commented that her experience
in the schools has taught her that mathematics is efficiency. Another
said that you can't have true efficiency without understanding. I
think that I have smart students. But anyway, that's not the point of
this blog.
One connection that I saw between the pitfalls paper and Dr.
Leatham's Pre-service teacher paper was this idea of "mastery first."
In Dr. Leatham's paper pre-service teachers had the belief that you
should master the mathematics before using technology to do the
mathematics. The pitfalls paper kind of had a different idea of
mastery, that you should learn everything about the software or
technology before you should think about using it to do mathematics. I
liked de Villiers' comment that you don't have to know everything
about a car's engine to be able to drive a car. However, I also don't
think that you should have people driving cars who know nothing about
how to drive the car. It can be dangerous and frustrating, kind of
like every time I try to use Maple for my PDE homework. I think de
Villiers gives some good ideas of how students with little knowledge
of the technology can still use it to explore and learn about both
mathematics and the technology at the same time.
I was grateful for de Villiers' argument that there is value in doing
certain things by hand in his "Makes Practice Obsolete" pitfall.
Naturally, I connected this to the Ralston paper on abolishing PPA.
Now I'm not sure if we ever quite decided what Ralston decided by PPA
or Mental Math, but my personal experience tells me that doing certain
things on your own with your own hands may have value, not in the
drill and practice sense, but a more kinesthetic sense. I appreciate
how de Villiers brought this up.
I think that we could also connect de Villiers' ideas about
misconceptions caused by the limited calculating capacities of
technology to the Paige, Seshayer, and Toda paper. And then there is
that visualization word that keeps coming up. I would like to know,
how do you guys define visualization? It's a term that we all keep
throwing around, but I'm not sure if we are talking about the same
thing. Erin at one point tried to help me understand what she may mean
by visualization, and I would love to resurrect that discussion if
anyone is interested.

[Rachelle]

I enjoyed de Villiers's papers, especially the one on the roles of
proof. I think I believed that there were so many different reasons
for proof, but I had never actually articulated them. It was nice to
read these ideas that jived well with my own, even though I had never
explicitly thought them before. I also wanted to mention that I really
love doing proofs for the intellectual challenge. In fact, I think
that's what I like about math and why I continued studying it.

But onto making connections. . . I thought the most obvious
connection between de Villiers's two papers was the "Insufficient
rethinking and evaluation" pitfall. In essence, his argument in the
roles of proof paper was that we need to rethink and expand our
concept of the role of proof. So many teachers believe and/or present
to students that the only role of proof is verification. This is what
makes it so frustrating for students when they are already completely
convinced of the truth of a proposition, especially when the proof
lends no further insight into why the proposition is true. De Villiers
argued that there are many more roles of proof which become important
when using dynamic geometry software and that introducing new
technology creates a need for rethinking and evaluating old classroom
norms/beliefs/practices.

I thought that this connection was strongly linked to the "no change"
pitfall and its connection to proof as verification. If teachers do
not change their classroom practices, then they are most likely not
rethinking, evaluating, and improving students' classroom experiences.
Also, if teachers keep verification as the only (or the most
important) kind of proof in their classroom, then they are not
changing according to the fundamental differences introduced by the
technology.

I also think the "visualization makes easier" pitfall connects to the
need for proofs as verification. De Villiers discussed how
verification type proofs are important in the context of dynamic
geometry (at least in part) because of accuracy issues. Even though
accuracy of the sketches is a helpful aspect of dynamic geometry, the
slight inaccuracy of the finite-decimal capacity of the measurement
tools creates a need for proofs to verify the truth of propositions.
Being able to "see" sketches that seem to support specific false
propositions is a pitfall if students are unwilling to take the next
step to prove and verify the seemingly correct proposition.

On Mar 20, 11:06 am, Keith <LethaLeat...@gmail.com> wrote:

[Rachelle]

Chris,
Way to go on connecting to past papers! I just stuck with the two de
Villiers papers but I really enjoyed reading all the connections you
found with everything else.

I've been thinking about what I mean when I think of "visualization."
I thought of one example, but I'm not sure that it really encompasses
all that I think visualization means. In calculus, we find the volume
of solids when you have a function and you rotate it around an axis. I
could never just remember the formulas for the different methods, but
I picture (in my mind) the curve, then rotating the curve around the
axis, and then the segment that I want to repeatedly add up so I know
what to take the integral of. As I'm doing this, I think I'm
visualizing the disc method, with some function like y=x^2 being
rotated around the x-axis. I picture what the solid looks like,
cutting the solid into discs, and then I know how to figure out the
area of those discs, and then I picture where the radius of each disc
could be found on the graph of y=x^2. I guess I think of visualization
as taking symbols or equations and thinking of the pictorial or
physical representations for the same ideas/objects. I'm pretty sure
most people would consider this visualization, so I'm guessing this
isn't the part of visualization that you're questioning.

If I were to define visualization as anything different or more broad
than that example, I really don't know what I would say. My curiosity
is piqued about what others think. Let me know if you think of
visualization in the same way I do or if you have a broader
interpretation! And Chris, what do you think?

> > have done thus far in the course.- Hide quoted text -
>
> - Show quoted text -

[Tenille]

I saw several connections among the roles of proof and the pitfalls of
dynamic geometry as discussed in the two de Villiars papers. The most
obvious to me was the role and pitfall of "verification". I noticed
others as well, but for purposes of time and brevity, I want to focus
on the second part of Keith's question.

I specifically want to discuss the connections I noticed between de
Villiars and Doerr and Zangor. Dynamic geometry software can be used
as a computational tool, and just as exclusive reliance on a
calculator can lead to false conclusions, dynamic geometry software
can also lead a student to false conclusions (as exemplified in the
verification sections). This illustrates why experiential evidence is
not adequate to convince mathematicians, and hence the need for
proof. This is probably why I would not make a good mathematician; I
am usually thoroughly convinced by GSP.

Dynamic geometry also serves as a transformational tool. De Villiars
stated that in "using dynamic geometry (or any other technology)
effectively, one really has to radically and critically rethink the
content, the aims, and the teaching approach one uses". I think that
dynamic geometry motivates the need for other roles of proof aside
from the role of verification (in fact, I usually only try to prove
things that I am already convinced are true).

I used GSP to collect data during one of my professional development
classes. I can't remember how or why, but I remember doing it. So I
guess dynamic geometry can be considered a data collection and
analysis tool.

Dynamic geometry also serves as a visualizing tool (although the
definition of visualization is still being negotiated in our class).
I liked how de Villiars pointed out that visualizing does not always
make proof easier. I often have a difficult time separating the
givens from what I want to prove when I am using GSP. I also think
that the visualization capabilities of GSP contribute to proof as a
means of discovery and explanation.

Finally, dynamic software can be used as a checking tool. I will
check my conjecture using GSP before I try to prove it. Again, I see
connections to proof as verification. Kudos to anyone who is still
reading this long response.

On Mar 20, 11:06 am, Keith <LethaLeat...@gmail.com> wrote:

[Janellie]

I enjoyed both articles we read for this week and would like to focus
mainly on dynamic software as it allows for exploration and
discovery. One of the pitfalls in de Villiers (2007) article was that
of "First Master Software." Many teachers think you need to teach the
students how to work with the dynamic software program fully before
they can explore and discover. In de Villiers (2002), he talks about
one of the roles of proof being discovery. In order to discover
mathematics, one does not need to know everything about the program.
Instead, they can use the drag feature to notice relationships of
ready made figures. In this way students can conjecture, explain, and
look for counter examples. If students had to master the technology
first, it would take up a lot of time, as well as lead to many faulty
constructions. The "construct dynamic figures" pitfall (de Villiers,
2007) captured this idea. Students need to know about the properties
of an object before they can construct one, and that requires
visualization and exploration.

When considering this role of proof, discovery, and the two pitfalls
above, I thought about Schattschneider and King's Preface: Making
Geometry Dynamic. They say dynamic geometry software is good for
accuracy of construction and exploration and discovery. Surely
constructions can be very accurate in places like geometer's
sketchpad, but we can not always expect students' constructions to be
accurate, especially if they do not have a chance to know a lot about
the software and the nature of the objects they are trying to create
before they construct. I like that deVilliers pointed the two above
pitfalls, because it was not adequate examined by Schattschneider and
King.

As far as exploration and discovery, Schattschneider and King note
that it would be less educative to just give students theorems and
definitions rather than have them discover important geometric
relationships. It would be better to let students test their
conjectures in a dynamic atmosphere. De Villiers (2002) states that
"To the working mathematician proof is therefore not merely a means of
verifying an already-discovered result, but often also a means of
exploring, analyzing, discovering and inventing new results" (p. 6).

One last connection I would like to point out. In Doerr and Zangor
(2000), one role of the graphing calculator is that it is a tool of
visualization. Also, Schattschneider and King discuss visualization
as something that dynamic geometry software is good for. However,
according to de Villiers (2007), visualization could prove to be a
pitfall. Students and teachers might rely too much on what they see
and be satisfied. This can lead to many faulty conjectures and
proofs, such as the ratios of the quadrilaterals discussed in de
Villiers (2002). Thus, teachers should be careful in leading their
students to realize the limitations of just relying on visualization.

[erin...@byu.edu]

Connections among roles of proof and pitfalls of dynamic geometry...
Hmmm...

Firstly, in the paper on roles of proof de Villiers (2002) quoted Paul
Halmos, who said "Efficiency is meaningless. Understanding is what
counts." I felt like this statement pretty much summed up the
stripped-bare, main ideas of these papers: that understanding is (or
should be) the driving force for all we do as mathematicians, whether
we use proof (as a "meaningful activity") or dynamic geometry
software, we need to be aware of the hindrances and helps (couldn't
come up with a better word) of the various roads we take toward that
understanding. de Villier also discussed the role/function of proof
as explanation, and in the geometry paper he stated his belief of "a
continued need for proof as the ultimate means of verification" -
proof/the software must do more than just show that something is true,
we need to show why. Proof as discovery reminds me of the "first
master software" pitfall in that the process of learning how to use
the software can help students develop mathematical ideas and can lend
itself to "discovery."

I also wanted to comment on a few things in the papers.
In the geometry paper I was reminded of an old article we read
(forgive me for leaving the old articles at home, maybe somebody else
knows the reference?) where the author claimed that technology drives
the skills... I found this evident in de Villiers' paper in the first
page when he said that "though new technologies will inevitably make
certain old skills obsolete, they will also require the development of
new skills." In the same paper, de Villiers quoted Sutherland who
said that we must "learn to use [new technology] in ways which
transform mathematical activity" - sounds pretty much like Doerr and
Zanger (2000) to me! Also in that paper de Villiers brings
visualisation into play during some of the pitfalls in that students
take what they see to be true and don't become conceptually engaged at
a higher level. A question that I had regarded de Villiers' statement
that one of the bottom lines to keep in mind with dynamic geometry
software is that it "was never intended to replace... important hands-
on activities" - I am skeptical here because I wonder then why it can
do everything that we can do by hand and more? Just as we see members
of society complain about simple algebra, wondering why they should
need to know how to do it if there are calculators that can do it for
them, I am curious as to whether this will become a problem with
constructions or other capabilities of such dynamic software. of
course this is not an issue at the moment because the use of such
software is not widespread. Then again, calculators were once upon a
time as unfamiliar as these programs... Just a thought.

In the proof paper de Villiers quoted Doug Hofstadter who said that
"direct [physical] contact [with constructions] is even more
convincing than a proof, because you really see it all happening right
before your eyes" - if such a thing is more convincing, why is there a
need for proof? My impression is that proof exists to convince us of
the truth of particular statements. And it can't just be a convinced
truth, it must be absolute truth (we are faulty humans and are easily
persuaded to believe many things that are not absolute truth). I
really thought it vital to discuss how proof is to help us
understanding not only that something IS true, but WHY it is so.
Later in the paper, de Villiers answers my most recent question (right
above the section of proof as a means of communication), but then I
wonder why he included the first statement? Do I have a different
interpretation of what "convince" means?

I absolutely loved when de Villiers discussed proof as a "testing
ground for the intellectual stamina and ingenuity of the
mathematician" and felt like I could climb a mountain. Not much else
to say about that part but I wanted you all to know I loved loved
loved it.

On Mar 20, 11:06 am, Keith <LethaLeat...@gmail.com> wrote:

[Shawn]

Before I went to write this post, I read through your answers to
Keith's question. I was so impressed with what you all had written
that I wasn't sure that I would be able to add to the discussion.
But, here goes:

First, under the pitfall of "Visualisation always makes easier," de
Villiers said that to the novice, dynamic geometry could be confusing
or distracting. I think that this is true with novices in proof
writing. Students seeing a full proof in geometry for the first time
might be confused seeing all those statements and reasons. They would
wonder why certain statements were chosen before others. I think the
solution to this phenomenon lies in the description of the pitfall of
"Construct Dynamic Figures." There he stated that students must first
explore the properties of figures like squares, rectangles, etc.,
before requiring the students to construct them with dynamic
geometry. I think that if the students knew the relationships among
and in certain figures, they will learn the steps to be able to prove
something is true and defend it with other students and the teacher.

A second connection between the papers is under the pitfall of
"Insufficient Rethinking & Evaluation." Here, de Villiers stated that
it is far more meaningful to introduce proof within a dynamic geometry
context. He mentions several of the roles of proof here, which are
proof as a means of explanation, understanding, and discovery. These
are exactly the roles mentioned in his article on the roles of proof.

Finally, I wanted to make a connection to Abramovich's paper on
Possible Learning Environments (PLEs). He used a spreadsheet and the
multiplication table as a PLE which was to help students go beyond the
traditional expectation for their learning. In the paper of the role
of function and proof, de Villiers cited a PLE (Van Aubel's Theorem in
Gardner, 1981) which was the connection of the midpoints from the
squares of the sides of a quadrilateral will make 90 degree angles
(see Figure 1). He, taking on the role of a student in a PLE in
dynamic geometry, extended the problem and looked at the connections
of the midpoints of rectangles constructed from the sides of the
quadrilateral. He was convinced that they made 90 degree angles, but
still needed to prove it. Hopefully that made a little sense.

On Mar 20, 11:06 am, Keith <LethaLeat...@gmail.com> wrote:

[Shawn]

Here is a stab at the meaning of visualization. I don't remember the
particular conversation, so this might be a little off from what you
all were saying. I think that visualization is being able to make a
mental picture of the mathematical topic spurred by a word. For
example, what comes to mind when you hear the word circle. Is it all
points that are equidistant from a center? Or is is just a perfectly
round figure? I think that an important thing in mathematics is that
there is a common image that comes to mind when one hears a certain
word. When one person says something about a circle, it is important
for them to know what a circle looks like in order to be able to talk
about that thing with relation to the circle. Sketchpad really helps
show what a circle is, from the time when you select the tool, fixing
a center by clicking and dragging the mouse out from there. This
shows that you are making a figure of all points equidistant from that
center.

On Apr 1, 1:04 pm, CJ <christine.johnson....@gmail.com> wrote:

[Tenille]

First, I want to add my two-cents to the discussion about defining
visualization. I liked Shawn's description of the mental images that
come to your head. The problem is that this differs from person to
person. What do you think of when I say the word apple? Your mental
picture is likely different from mine. Similarly, when I think of
circle, I actually picture a unit circle on the coordinate plane with
its center at the origin accompanied by its equation. This brings up
an interesting question: can thinking of an algebraic equation be
considered visualization. I think for some people (not me) it might.

Erin, I enjoyed your thoughts about whether or not constructions
should still be down "by hand". I think that Chris also discussed the
importance of "hands on" activities, especially in a kinesthetic
sense. I have a question: when would the need to be able to construct
something ever be used outside of geometry class. I honestly don't
know. Do carpenters and construction workers use the properties of
shapes to construct them? I guess I am wondering because Erin seemed
to suggest that one day dynamic geometry software could be as wide
spread as calculators. Do you think that GSP will ever be used as
more than an educational tool? I think that some of you may disagree
with this next statement, but I would consider constructions in GSP to
be very "hands on". This reminds me of the article about physical and
virtual manipulatives. I have constructed the nine-point circle now
in three different classes: once by hand and twice using GSP. I am
not claiming that constructing the nine-point circle by hand was the
same type of hands on activity as constructing it using GSP; in fact,
they are completely different to me. When I constructed the nine-
point circle by hand, I felt very bogged down by trying to be
accurate, not to mention all the extra arcs and segments. I honestly
didn't focus on the mathematics on the problem (for more reasons than
just trying to be accurate). However, I felt that the clean
construction in GSP helped me to understand the construction better.
What are some benefits to paper and pencil constructions?

A few more thoughts that aren't related to anyone's blog in
particular. I think that not all proofs fulfill all the different
roles of proof. In order to understand these different roles better,
I think it might be nice to look at different proofs that exemplify
these different roles. I'll bring examples to class and we can
discuss them if anyone else so desires (but if you don't want to,
that's fine). Also, I think that some roles of proof are more
important depending on the situation. For example, as a high school
geometry teacher, I consider the role of verification to be the least
important (and ironically the most universally given) because my
students may not view verification as important and may not find some
proofs very convincing. However, as a mathematician, I think that the
role of verification is supremely important. Anyway, just some
thoughts. By the way, does anyone read the blog posts after they post
their 2nd post? I always try to read them just before class, but I
wonder if the people that have already finished their blogging will
read this.

[Tenille]

I forgot to reference von Glassersfeld (I hope I spelled that right)
in the apple example. I'm pretty sure that he used that example in
the paper we read in 590, but it could have just been part of our
discussion and not from his paper. But just in case, I thought that I
had better give credit where credit is due.

[CJ]

This is kind of reply to Tenille and kind of an extension of the whole
visualization discussion. I think that visualization for me is anytime
I see something in my head that can be seen in real life. Yep, that's
confusing. But if I've seen an equation before and I can see it in my
head when I need it, I think that is a form of visualization. But then
it makes me wonder what visualization is like for someone who has
never "seen" anything (aka someone who was born blind). Would they
consider memories of smells, motion, texture a type of visualization?
I think that Rachelle and Shawn's ideas of visualization fit into
mine. I also think of the statement that "if you draw a picture you
will understand it better," which I realize is not always true, but I
think deep down I believe that drawing "the right" picture always will
help in understanding. Visualization, I think, is one of my
mathematical biases, because it is such an integral part of my
understanding of mathematics that it really is hard for me to relate
to or believe someone when they say that drawing a picture doesn't
help.
Which brings me to de Villiers. I was quite disappointed in the
evidence that he provided for his "visualization always makes easier"
pitfall. First of all, he is pretty vague about "the task" that the
students did with and without Logo. It says it involved a proof, but
what kind of proof? A "manipulate-the-symbols" syntactical proof? Or a
proof that helps you to understand why (some call this a semantic
proof)? Did the "better" students do better because they received the
problem in a context-less frame where they didn't even have to
consider other variables that might be relevant? If so, does this
really prepare them for real life? As I look around this world, "data
and visual overload" seems to be the way we are heading. Maybe it is
valuable to learn to solve problems (including identifying what is
important) in these environments. Then again, maybe not.

[erin...@byu.edu]

I read the posts before class! But I agree with you in that not
everybody else does...

Like last time, i have a few responses for a few people:
Chris, touche on your comments! i love that you pointed out the whole
chicken/egg thing with this mastery of technology before/after mastery
of content knowledge involved. I agree with you in that the basics of
both are essential but that there's a lot of learning/growth that can
occur with even just the basics.

On Merriam-Webster.com, they define visualization as: 1 : formation of
mental visual images
2 : the act or process of interpreting in visual terms or of putting
into visible form

so I looked up what "visual" means and it came up with:1 : of,
relating to, or used in vision <visual organs>
2 : attained or maintained by sight <visual impressions>
3 : visible <visual objects>
4 : producing mental images : vivid
5 : done or executed by sight only <visual navigation>
6 : of, relating to, or employing visual aids

So I think what Rachelle and Shawn and chris (and any of the rest of
you) were saying about it are pretty much on par.

With regards to visualization, Tenille, you said that the
visualization capabilities of GSP contribute to proof as a means of
discovery and explanation. How? I feel like that is still kind of
mysterious to me and I don't love feeling like I'm in the dark.
(Tenille, i read all of your long response, so can I have a kudo? I
like the peanut butter ones best).

Rachelle - you said that there are "many more roles of proof which
become important when using dynamic geometry software" - really?
Which ones? I'd love to see you make more of that connection... Not
that I disagree, but I'd like to know more of your thought process.

Kind of along those same lines Janelle, you said we use proof as
discovery in dynamic geometry software... This also flies over my
head - I understand the discovery part, but not how proof can be used
as discovery in this software? I feel like my brain almost agrees
with you, but if it were articulated in greater depth then I think I
would be happier. Also, you said something about visualization that I
disagree with - I feel like you said that all of these papers imply
that we should rely only on visualization for learning... but
everything I've ever read has, I feel, encouraged visualization as
supplemental and not as a primary method for students to use to build
understanding. Am i wrong or crazy?

On Apr 3, 8:41 am, Tenille <tenille.can...@gmail.com> wrote:

[Janellie]

I'm not sure who I'm replying to anymore. I like what everyone has
said about visualization. For me, it is not just a picture. I don't
have to be able to draw a picture in order to visualize something. It
may not have to be a picture, but it is something that I can "see".
It is not always something I've seen with my eyes before either. I
can try to visualize what is going to happen in the future. However,
I use what is already in my experience to help me develop my
visualization. Therefore, those people who are blind, in my
definition, would be able to visualize things as well. They can "see"
things in their mind. Visualizing is connecting things you encounter
to things you have experienced before. However, I don't think most
people see visualization the way I do. In the papers, I think
visualization is being able to see a picture or diagram of something
and think of how it connects to the mathematics. I don't know how
much sense this paragraph makes, but there it is.

A question that Erin asked, I think, relates to this discussion. She
stated (quoting the article): software "was never intended to

replace... important hands-
on activities" - I am skeptical here because I wonder then why it can
do everything that we can do by hand and more?

I don't think I have the answer to that question, but I have some
thoughts. Sometimes, the software is insufficient because it can't
help us visualize in the way we need to in order to do the needed
mathematics. Or, it helps us visualize things, but we might need to
see it in a different way. Also, it might be easier to use
manipulatives that you can alter physically than to have to make a
substitute on the computer.

[Shawn]

I try to read them just before class, too. Most of the time I'm
successful.

Just wanted to say a word about your apple analogy, which was very
good. I think that sometimes in math we do say a word like apple and
people get different images. I think that one of the goals of
mathematics is to get people familiar enough with the subject that
when someone says apple, or circle, they can visualize something
somewhat similar that they can communicate meaningfuly about. Thanks
for pointing the out the possible contradiction with your apple
example.

On Apr 3, 8:41 am, Tenille <tenille.can...@gmail.com> wrote: